Logic for Knowledge Representation

1. Logic for Knowledge Representation Sindhu Kutty

2. Why Knowledge Representation? • What else is there?? • Lack of motivation for ‘reinventing the wheel’! • Declarative semantics

3. Logical AI • An agent’s knowledge of its world and goals represented by sentences in logic • Sound and complete inference rules lead to useful deductions • Epistemological and heuristic problems

4. Classical Logic • Logic of truth • Operators • ,,, • (ab) (ab) • Quantifiers • , 

5. Some examples • Everyone has a father childman father(man,child) • Everyone loves someone personsomeone loves(person,someone) • There is someone who everyone loves someoneperson loves(person,someone)

6. The truth about implication • The truth table • apple(x)fruit(x) Am I a liar if ‘x’ happens not to be an apple?

7. Classical Logic - Issues • Truth as opposed to knowledge • What does yx father(x,y) mean? • Is that enough?

8. Computability Logic - Motivation • The difference between • yx father(x,y) • ㄇyㄩx father(x,y)

9. Computability Logic • Logic of computability • Logic of interaction • Resource conscious • Some very expressive fragments are recursively enumerable • It captures constructive ability as distinct from truth

10. Semantics • Computational problems understood as games played by a machine against the environment • Representation:  - environment ㄒ - machine

11. Elementary Games • Move • An observable action by an agent • Run • Sequence of moves • Legal runs • Even(5) • Who wins this? • What are the moves? • What is the run? Legal runs?

12. Elementary Games • Even(5)  Even(4) • Who wins this? • What are the moves? • What is the run? Legal runs? • (Even(5)  Even(4)) • Who wins this? Role reversal • What are the moves? • What is the run? Legal runs?

13. Constant Games • Even(5) ㄩ Even(4) • Choice disjunction • Who wins this? • What are the moves? • What is the run? Legal runs? • (Even(5) ㄇ Even(4)) • Choice conjunction • Who wins this? • What are the moves? • What is the run? Legal runs?

14. Games • Even(x) ㄩ Even(x) • Who wins this? • What are the moves? • What is the run? Legal runs?

15. Games • A game is a function from valuations to constant games • A valuation is a function from the set of variables to the set of constants

16. Games • A = Even(x) ㄩ Even(x) • A(x/5) • Who wins this? • What are the moves? • What is the run? Legal runs?

17. Choice Existential Quantifier • A = ㄩx (Even(x) ㄩ Even(x)) • <ㄒ5>(ㄩx (Even(x) ㄩ Even(x))) • <ㄒ2>( Even(5) ㄩ Even(5)) • Run • <ㄒ5, ㄒ2 > • ㄒ- won run

18. Choice Universal Quantifier • A = ㄇx (Even(x) ㄩ Even(x)) • < 5>(ㄇx (Even(x) ㄩ Even(x))) • <ㄒ2>( Even(5) ㄩ Even(5)) • Run • < 5, ㄒ2 > • ㄒ- won run • Run • <> • ㄒ- won run

19. Blind Quantifiers • A = x (Even(x) ㄩEven(x)) • What run makes this ㄒ- won ? • A = x (Even(x) ㄩEven(x)) • What run makes this ㄒ- won ?

20. Games • ㄇx ((Odd(x) ㄩ Odd(x)) → (Even(x) ㄩ Even(x))) • Who wins this? • Resource •  x (Even(x) ㄩOdd(x) → ㄇy(Even(x*y) ㄩ Odd(x *y))) • Can this be won?

21. Interesting Observations! • A A is valid. Why? • For A = Even(5) • A ㄩA is not valid. Why? • The difference between • yx father(x,y) • ㄇyㄩx father(x,y)

22. CL4 • Operators used so far • CL4├ F iff F is valid • Uniform constructive soundness • Strong completeness • F* not computable for some interpretation

23. An example • ㄩxㄇy Has(x,y) • ㄇx(ㄩs Symptoms(x,s)  ㄇt (Positive(x,t)ㄩ Positive(x,t))  ㄩy Has(x,y)) • How can ㄒ win this game?

24. Parallel Recurrence • Parallel Recurrence • A  A  A  … or A • When does ㄒ win this?

25. Branching vs. Parallel Recurrence • Branching Recurrence • Legal run of A can be thought of as a tree where each branch spells a legal run • ㄒ wins iff it wins in all branches • Root is the empty run • Only  has the capability of making a splitting move - easier for  to win - why? • Which is easier for  to win: • A or A? Why?

26. Some Examples • Potential knowledge of everyone’s gender • ㄇx(Female(x) ㄩ Female(x)) • Any difference if we say • ㄇx(Female(x) ㄩ Female(x))

27. Back to the Game… • ㄇx(ㄩs Symptoms(x,s)  ㄇt (Positive(x,t)ㄩ Positive(x,t))  ㄩy Has(x,y))

28. Resource Consciousness • What if n tests were needed? • -conjunction of n identical conjuncts • What if an unbounded number of tests was needed? • ㄇt (Positive(x,t)ㄩ Positive(x,t))

29. Another Example • x (Red(x) Acid(x)) • x (Acid(x)  Red(x)) • ㄇx(Red(x)ㄩRed(x)) • Does KB  ㄇx(Acid(x)ㄩAcid(x))?

30. Possible Worlds Analysis • TRUE(KNOW(A,IMP(ACID,RES(DO(A,TEST),AND(ACID,RED))))) • TRUE(KNOW(A,IMP(NOT(ACID),RES(DO(A,TEST),AND(NOT(ACID),NOT(RED)))))) • TRUE(ACID) • Prove: TRUE(RES(DO(A,TEST),KNOW(A,ACID)))

31. Computability Logic • Epistemic variants of classical logic get messy • They are also non-semidecidable • The KNOW operator is not constructive

32. An Epistemic Variant • Axioms • M1 P, s.t. P is an axiom of ordinary propositional logic • M2 KNOW(A,P)  P (knowledge axiom) • M3 KNOW(A,P)  KNOW(A,KNOW(A,P)) (positive introspection axiom) • M4 KNOW(A,(PQ))  (KNOW(A,P)  KNOW(A,Q)) (logical omniscience) • M5 If P is an axiom, then KNOW(A,P) is an axiom

33. Possible World Analysis • Individual propositions known by an agent • State of affairs compatible with what agent knows • Model theory

34. Accessibility Relation • K(A, W1, W2) Possible world W2 is compatible or consistent with what A knows in possible world W1 • For the following illustrations Agent A knows PW0 is the actual world • Goal: Capture axioms M1-M5 model-theoretically

35. Accessibility Relation - Definition • If agent A knows P in W0, then P is true in every accessible world P KA W1 P KA P W0 W2 KA P Wn

36. Accessibility Relation - Properties • M2 KNOW(A,P)  P • If agent A knows P, then P is true a1 K(a1, W0, W0) KA P W0

37. Accessibility Relation - Properties • M4 KNOW(A,(PQ)) (KNOW(A,P)  KNOW(A,Q)) • If P is true in every world W1 s.t. K(A,W0,W1) then A knows that P in the actual world P KA KA W1 P KA P W2 W0 KA P Wn

38. Accessibility Relation - Properties • M3 KNOW(A,P)  KNOW(A,KNOW(A,P)) • a1 w1 w2 (K(a1, W0, w1)  (K(a1, w1, w2)  K(a1, W0, w2))) KA P KA P KA KA P KA KA P KA KA W0 P KA P KA

39. Accessibility Relation - Properties • M5 If P is an axiom, then KNOW(A,P) is an axiom • Generalize these constraints to hold not just for the actual world but for all possible worlds and for all agents

40. The Example Revisited • x (Red(x) Acid(x)) • x (Acid(x)  Red(x)) • ㄇx(Red(x)ㄩRed(x)) • Then CL4├ KB  ㄇx(Acid(x)ㄩAcid(x))

41. The Example Revisited • TRUE(KNOW(A,IMP(ACID,RES(DO(A,TEST),AND(ACID,RED))))) • TRUE(KNOW(A,IMP(NOT(ACID),RES(DO(A,TEST),AND(NOT(ACID),NOT(RED)))))) • TRUE(ACID) • Prove: TRUE(RES(DO(A,TEST),KNOW(A,ACID)))

42. To Sum Up • Classical Logic • Querying unnatural • Epistemic variants • not constructive • messy • Advantages of Computability Logic • Simplicity • Resource consciousness • Uniform Constructive Soundness of CL4

43. References • Japaridze, G. Introduction to computability logic. Annals of Pure and Applied Logic, 123 (2003), pp.1-99. • Japaridze, G. Computability logic: a formal theory of interaction, 2004. • Moore, R. A formal theory of knowledge and action, Formal Theories of Commonsense Worlds(1985), pp. 319-358. • McCarthy, J. Concepts of Logical AI, in progress.